Centre and Spread of Data: Quick Revision

A brief note on Centre and Spread of Data. Read it before a test. Try each question at the end before you open its answer.

The three averages

• Mean: $\bar{x} = \dfrac{\sum f x}{N}$. For grouped data use the class marks.
• Median: the middle value of the ordered data. For grouped data, $l + \dfrac{\frac{N}{2} - C}{f}\,h$.
• Mode: the most common value. For grouped data, $l + \dfrac{f_1 - f_0}{2f_1 - f_0 - f_2}\,h$.
• Rule of thumb: $\text{mode} \approx 3\,\text{median} - 2\,\text{mean}$ for slightly skewed data.
• Step deviation: with $u = \dfrac{x - A}{h}$, the mean is $A + h\,\bar{u}$.
• Choice: an extreme value pulls the mean. The median is steady.

Measures of spread

• Range: largest $-$ smallest. Coefficient of range: $\dfrac{\text{largest} - \text{smallest}}{\text{largest} + \text{smallest}}$.
• Mean deviation about $a$: $\dfrac{\sum f \lvert x - a \rvert}{N}$. It is smallest when $a$ is the median.
• Variance: $\sigma^2 = \dfrac{\sum f (x - \bar{x})^2}{N} = \dfrac{\sum f x^2}{N} - \bar{x}^2$.
• Standard deviation: $\sigma = \sqrt{\sigma^2}$, in the unit of the data.
• Step deviation: $\sigma^2 = h^2\left[\dfrac{\sum f u^2}{N} - \left(\dfrac{\sum f u}{N}\right)^2\right]$.
• Sample: A-Level, IB and AP divide by $n - 1$ to estimate the population variance.

A worked example

Find the mean, the median, the variance and the standard deviation of $3, 8, 5, 10, 14$.

1. The mean. $\bar{x} = \dfrac{40}{5} = 8$.
2. The median. The ordered values are $3, 5, 8, 10, 14$, so the median is $8$.
3. The squared distances. $25, 0, 9, 4, 36$, which add to $74$.
4. The variance. $\sigma^2 = \dfrac{74}{5} = 14.8$, and $\sigma = \sqrt{14.8} = 3.85$.
5. As a sample. $s^2 = \dfrac{74}{4} = 18.5$.

Practice with new numbers

A program to try

The program finds the mean, the median, the variance and the standard deviation of the list in the worked example. It also prints the sample variance.

Mistakes to avoid

• Dividing by the number of classes. Divide by $N$.
• Dropping the absolute value in the mean deviation.
• Forgetting the square root for the standard deviation.
• Forgetting $h^2$ in the step-deviation variance.
• Using $n$ for a sample. A question that says sample wants $n - 1$.

Check yourself

1. Find the mean and the median of $6, 9, 4, 12, 9$.

2. Find the range and the coefficient of range of $10, 25, 18, 40$.

3. Find the mean deviation about the mean of $4, 6, 8, 10, 12$.

4. For $10$ values, $\sum x = 50$ and $\sum x^2 = 300$. Find the standard deviation.

5. Which average would you quote for incomes, when a few are very large, and why?

Sources

• OpenStax, Introductory Statistics, Chapter 2, sections on measures of the centre and the spread of the data.
• NCERT, Mathematics for Class XI, Chapter 13 (Statistics).